Final answer:
To find the point on the curve y = x^2 where the tangent line has the greatest slope, calculate the derivative of the curve equation and evaluate it at different points. The greatest slope occurs at x = 1 on the curve.
Step-by-step explanation:
The slope of a curve is equal to the slope of the tangent line at a specific point on the curve. To find the point on the curve y = x^2 where the tangent line has the greatest slope, we need to determine the slope of the tangent line at different points on the curve. Since the curve is a parabola, the slope of the tangent line changes at different points. We can find the greatest slope by calculating the derivative of the curve equation and evaluating it at different points.
The derivative of y = x^2 is y' = 2x. To find the slope of the tangent line at a specific point, we plug the x-coordinate of that point into the derivative equation. Let's evaluate the derivative at two points: x = 1 and x = -1.
At x = 1, the derivative is y'(1) = 2(1) = 2. This means that the slope of the tangent line at the point (1, 1) is 2.
At x = -1, the derivative is y'(-1) = 2(-1) = -2. This means that the slope of the tangent line at the point (-1, 1) is -2.
Comparing the slopes, we can see that the tangent line has the greatest slope at x = 1 on the curve y = x^2.